Asymptotic Behaviour and Correctors for Dirichlet Problems in Perforated

ANNALIDELLA SCUOLA NORMALE SUPERIOREDI PISA Classe di Scienze

GIANNI DAL MASO FRANÇOIS MURAT Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 24, no 2 (1997), p. 239-290

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Asymptotic Behaviour and Correctors for Dirichlet Problems in Perforated Domains with Homogeneous Monotone Operators

GIANNI DAL MASO - FRANCOIS MURAT

0. - Introduction

In this paper we study the asymptotic behaviour of the solutions of some nonlinear elliptic equations of monotone type with Dirichlet boundary conditions in perforated domains. Let S2 be a bounded open set in e, and let A: Ho ’ p ( S2 ) ~ H -1’ q ( SZ ) be a monotone operator of the form

where 1 p +cxJ, 1 / p + 1, and a : --~ kv satisfies the classical hypotheses: Carath6odory conditions, local Holder continuity, and strong monotonicity (see (1.4)-(1.7) below). We suppose, in addition, that a (x, -) is odd and positively homogeneous of degree p - 1. Given a sequence of open sets contained in S2, we consider for every f E the sequence (un ) of the solutions of the nonlinear Dirichlet problems

We extend un to Q by setting un - 0 on and we consider (un ) as a sequence in The problem is to describe the asymptotic behaviour of the sequence (un ) as n tends to infinity. The main theorem of the paper is the following compactness result (The- orem 6.6), which holds without any assumption on the sets Qn. For every sequence of open sets contained in Q there exist a subsequence, still denoted by (Qn), and a non-negative measure it in the class such that for every f E the solutions Un of (o.1) converge weakly in to the solution u of the problem

Pervenuto alla Redazione il 21 marzo 1996. 240

Here ~, ~ ) denotes the duality pairing between H-1 ~q (SZ) and Ho ’ p (S2), while is the class of all non-negative Borel measures on Q with values in [0, +oo] which vanish on all sets of (1, p)-capacity zero. Moreover, we prove (Theorem 6.8) that (un) converges to u strongly in for every r p, and that (a (x, D u n ) ) converges to a (x, Du) weakly in RV) and strongly in RV) for every s q. Finally the energy (a (x, DUn), D u n ) converges to (a (x, Du), Du) + weakly* in the sense of Radon measures on Q. To prove this compactness result we observe (Remark 2.4) that all problems of the form (o.1 ) can be written as problems of the form (0.2) with a suitable choice of the measure it = in the class Actually we prove (Theorem 6.5) that, for every sequence (pn) of measures of the class there exist a subsequence, still denoted by and a measure /,I E such that for every f E H-1,q (Q) the solutions un of the problems

converge weakly in ~o’~(~) to the solution u of (0.2).

In our proof an essential role is played by the solution wn of the Dirichlet problem or, in the general case (0.3), by the solution wn of the problem

As the sequences (un) and (wn) of the solutions of (0.3) and (0.4) are bounded in (Theorem 2.1), we may assume that (un) and (wn) converge weakly in H6’ (Q) to some functions u and w. By using a result of [4] we prove that and converge to D u and D w strongly in for every r p (Theorem 2.11). The crucial step in the proof of our compactness theorem is the following corrector result (Theorem 3.1), which is interesting in itself: if f E then 241 where

and

The idea of the proof is to consider the function which, due to the homogeneity of a(x, ç), satisfies an equation similar to (0.3). We then use un - uwn as test function in the difference of (0.3) and of this equation. Using the strong monotonicity of a (x , ~ ) we prove that (u n - tends to 0 strongly in H¿’P (r2) (Theorem 3.7). Actually the previous description of the proof is not accurate, because in general w can vanish on a set of positive Lebesgue measure. For this reason we are forced to replace uwn by w~~ with 8 > 0, and to consider separately the sets where w (Lemmas 3.4, 3.5, 3.6). Another important step in our proof is the construction of the measure it, which depends only on w and on the operator A. We prove (Section 5) that v = 12013Au; is a non-negative Radon measure on Q which belongs to H-1,q (2), and we define the measure It by

where Cp (E) is the (1, p)-capacity of E with respect to Q. We prove also that this measure to (Theorem ’ belongs 5.1). It remains to prove that u coincides with the solution of (0.2) corresponding to the measure it defined by (0.6). To give an idea of this part of the proof, we assume now that 2 p (the case 1 p 2 is more technical) and that f E (the case f ~ needs a further approximation). Given w E Co (S2), we take v = as test function in (0.3). We would like to take v = as test function in (0.4). This is not always possible, thus as before we replace w by w v e for some E > 0 and we take v - test function in (0.4). Subtracting the two equations we obtain

I i n-2

Using (0.5) and the homogeneity of A, as well as some technical lemmas proved in Section 4, we pass to the limit first as n oo and then as s 0, obtaining 242

As by definition v = 1 - A w, we get

and thus (0.6) gives

Since the set (p E is dense in n (Proposi- tion 5.5), it follows that u is the solution of (0.2). In the case 1 p 2 the main difficulty lies in the fact that the function jUlp-2 u does not belong to for every u E We solve this problem by considering a suitable locally Lipschitz modification, nepr the origin, of the function t « which enables us to complete the proof following the ideas of the case 2 p +oo.

When p = 2 and A is a symmetric linear elliptic operator, the compactness result proved in the present paper has already been obtained by r-convergence techniques in [2], [1], [23], [7], [39]. These results were recently extended to the vectorial case by [27]. When 1 p +00 and A is the subdifferential of a convex functional = Ig D u ) d x defined on ~/o~(~), with ~/r~ (x , ~ ) even and positively homogeneous of degree p, the compactness result was proved in [20] by r-convergence techniques. The general case .) is not homogeneous can be obtained by putting together the results of [2], [15], [16] on obstacle problems. Further reference on this subject can be found in the book [18] and in the recent paper [21], which contains a wide bibliography on the linear case. Another method used in the study of this type of problems was introduced in [31] and [37], where the asymptotic behaviour of the solutions of (0.1) in the linear case was investigated in various interesting situations, under some assumptions involving the geometry or the capacity of the closed sets The case of general Leray-Lions operators in T~’~(~), with A possibly non homogeneous, was treated in [40]-[45], [32], [26] under similar assumptions. The linear case was also investigated in [14] by using test functions as in the energy method introduced by Tartar [46] in the study of homogenization problems for elliptic operators. In this paper some restrictive hypotheses on the sets S2n were made in terms of the test functions, and a corrector result was obtained. The same method was then extended to a more general situation including also the case where 1 p +oo and A is the p-Laplacian (see [2], [33], [38], [11]). The linear problem with a perturbation with quadratic growth with respect to the gradient was recently solved in [8] and [9]. A new decisive step was achieved in [21], where a compactness result without any hypothesis on the sets was proved in the (not necessarily 243 symmetric) linear case by introducing a new sequence of test functions which were defined as the solutions of equations (0.4) above with p = 2. Our paper follows the same lines as far as the test functions are concerned, but presents an alternative method of proof for the compactness and corrector results. In conclusion, the compactness result proved in the present paper is new when the homogeneous monotone operator A is nonlinear and is not the subdifferential of a convex functional. The corrector result, the strong convergence of the solutions un in for r p, and the convergence of the fields a (x, D u n ) are new for all nonlinear homogeneous monotone operators, including the case of subdifferentials of convex functionals: they are indeed obtained without any hypothesis on the sets An expository version of the present work, restricted to the case p = 2 and including also a simplified, but almost self-contained version of the proofs, has been published in [25]. The results of the present paper have recently been extended to the case of a monotone operator without homogeneity conditions in [10] and in [12], where the case of systems is also solved.

1. - Notation and preliminary results

Sobolev spaces and capacity. Let us fix a bounded open subset Q of lkv and two real numbers p and q, with 1 p +cxJ, 1 q +oo, and 1 /q + lip = 1. The space of distributions in Q is the dual of the space The space is the closure of Co (SZ) in the Sobolev space H 1’p (S2), and the space H-1’q (S2) is the dual of On we consider the norm

and is endowed with the corresponding dual norm. For every subset E of Q the (1, p)-capacity of E in Q, denoted by Cp (E), is defined as the infimum of fo I D u I P dx over the set of all functions u E such that u > 1 a.e. in a neighbourhood of E. We say that a property P(x) holds quasi everywhere (abbreviated as q.e.) in a set E if it holds for all x E E except for a subset N of E with Cp (N) = 0. The expression almost everywhere (abbreviated as a.e.) refers, as usual, to the Lebesgue measure. A function u: S2 ~ R is said to be quasi continuous if for every s > 0 there exists a set A c Q, with Cp (A) s, such that the restriction of u to is continuous. It is well known that every u E has a quasi continuous representative, which is uniquely defined up to a set of capacity zero. In the sequel we shall always identify u with its quasi continuous representative, so that the 244 pointwise values of a function u E are defined quasi everywhere in S2. We recall that, if a sequence (un) converges to u strongly in then a subsequence of (un ) converges to u q.e. in S2. For all these properties of quasi continuous representatives of Sobolev functions we refer to [47], Section’3, and [30], Section 4. A subset U of Q is said to be quasi open if for every 8 > 0 there exists an open subset V of S2, with Cp(V) 8, such that U U V is open. We shall frequently use the following lemma about the approximation of the characteristic function of a quasi open set. We recall that the characteristic of a set is defined = if x E and function 1 E E C Q by 1,1 E, by

LEMMA 1.1. For every quasi open subset U of S2 there exists an increasing sequence (vn) of non-negative functions of which converges to 1 U quasi everywhere in S2.

PROOF. See [16], Lemma 1.5, or [21], Lemma 2.1. D

Measures. By a Radon measure on S2 we mean a continuous linear functional on the space of all continuous functions with compact support in S2.. It is well known that for every Radon measure À there exists a countably additive set function tt, defined on the family of all relatively compact Borel subsets of S2, such that ~, (u ) = for every u E Cho(0.). We shall always identify the functional X with the set function it. .. By a non-negative Borel measure on S2 we mean a non-negative, countably . additive set function defined in the Borel a-field of Q with values in [0, -f-oo]. It is well known that every non-negative Borel measure which is finite on all compact subsets of S2 is a non-negative Radon measure. We shall always identify a non-negative Borel measure with its completion (see, e.g., [28], Section 13). If it is a non-negative Borel measure on S2, we shall use L~ (S2), 1 r +oo, to denote the usual Lebesgue space with respect to the measure tt. We adopt the standard notation when it is the Lebesgue measure. We denote by Mg (0.) the set of all non-negative Borel measures p on Q such that

(i) = 0 for every Borel set B C Q with Cp(B) = 0, (ii) A(B) = U quasi open , B c U } for every Borel set B C S2 . Property (ii) is a weak regularity property of the measure p. Since any quasi open set differs from a Borel set by a set of Cp-capacity zero, every quasi open set is ti-measurable for every non-negative Borel measure it which satisfies (i). Therefore Jvt(U) is well defined when U is quasi open, and condition (ii) makes sense. REMARK 1.2. Our class coincides with the class M;(0.) introduced in [20]. It is slightly different from the classes .Mp and considered in [19] and [36], where condition (ii) is not present. It is well known that every non-negative Radon measure satisfies (ii), while there are examples of 245

non-negative Borel measures which satisfy (i), but do not satisfy (ii). Condition (ii) will be essential in the uniqueness result given by Lemma 5.4, which is used in the proof of the uniqueness of the yA-limit (Remark 6.4). 0 For every open set U c Q we consider the Borel measure pu defined by

As U is open, it is easy to see that this measure belongs to the class The measures / will be useful in the study the asymptotic behaviour of sequences of Dirichlet problems in varying domains (see Remark 2.4 and The- orem 6.6). E .J~lo (S2), then the space n L§(Q) is well defined, since all functions in are defined ~-almost everywhere in Q. It is easy to see that is a Banach space with the norm = r o p [5], Proposition 2.1 ). ~ (Q) I-L 0B,> Finally, we say that a Radon measure v on S2 belongs to H-1’q (S2) if there exists f e H -1’ q ( S2 ) such that

where (.,.) denotes the duality pairing between H -1 ~ q ( S2 ) and Ho ’ p ( S2 ) . We shall always identify f and v. Note that, by the Riesz theorem, for every non- negative functional f E H-1’q (S2) there exists a non-negative Radon measure v such that (1.2) holds. It is well known that every non-negative Radon measure which belongs to H-1’q (Q) belongs also to The monotone operator. Let a: Re be a Borel function satisfying the following homogeneity condition:

for every x E 0, for every t E R, and for every ~ E with the convention ltlp-2 t = 0 for t = 0 and 1 p 2. In the case 2 p we assume that there exist two constants co > 0 and cl > 0 such that

for every x E Q and for every ~1, ~2 E e, where (~, ~) denotes the scalar product in 1~. In the case 1 p 2 we assume that there exist two constants co > 0 and c1 > 0 such that 246

for every x E S2 and for every ~l , ~2 E e with ~1 ~ ~2. In particular, (1.3) implies that

for every x E S2 and for every ~ E Rv, hence

for every x E S2, while (1.4)-(1.7) and (1.9) imply that

for every x E Q and for every ~ E We now define the operator A: H-1’q (S2) by Au - -div(a (x, Du)), i.e.,

for every u E and for every v E H6" P (0). This operator is strongly monotone on Ho ’ p ( SZ) . The model case is the which corresponds to the choice a (x , ~ ) _ ~ ~ ~ p-2 ~ . Conditions (1.4)-(1.7) are satisfied in this case, since

for 2 p +oo, while

for 1 p s 2 and §1 ~ ~2. 247

2. - Relaxed Dirichlet problems

Estimates for the solutions. Let JL e .J~lo (S2), f E H-l~q (S2), and let A be the operator defined by (1.12). We shall consider the following relaxed Dirichlet problem (see [22] and [23]): find u such that

The name "relaxed Dirichlet problem" is motivated by Theorem 6.6 and is generally adopted for this kind of problems (see also the density results proved in [23] and [22]). More generally, given 1/1 E we shall consider the following problem: find u such that

The existence of a solution of (2.2) is given by the following theorem. THEOREM 2.1. Let it E 1/1 E H1,P(Q) n L~ (S2), and f E H-1,q(Q). Then problem (2.2) has a unique solution. Moreover the solution u of (2.2) satisfies the estimate

where C is a constant depending only on p, co, ci. PROOF. Let B : Ho ’ p ( S2 ) f 1 L ~ ( S2 ) -~ ~ Ho ’ p ( S2 ) n L ~ ( S2 ) ) ~ be the operator defined by

for every z, v L~ (Q). It is easy to see that B is monotone, continuous, and coercive. As f can be identified with an element of there exists a solution Z E n L~ (S2) of the equation Bz = f in (see, e.g., [35], Chapter 2, Theorem 2.1). It is clear that u - z + 1/1 is a solution of (2.2). The uniqueness follows easily from the monotonicity conditions (1.4), (1.6), (1.13), and (1.14) by a standard argument. Let us prove the estimate (2.3). If we take v = u - 1/1 as test function in (2.2), we obtain 248

Therefore, by the coerciveness condition (1.10) and by the boundedness condition ( 1.11 ) we have

which implies (2.3) by Young’s inequality. 0

The following lemma will be used to prove the continuous dependence on f of the solution of (2.2).

LEMMA 2.2. Let it E let u 1, U2 E H1,P(Q) let ~p E rl L °° ( S2) with cp > 0 q.e. in Q. If 2 p +oo, then

If 1 p 2, then

where K (u 1, U2, ~p) stands for

PROOF. If 2 p (2.5) follows from the monotonicity conditions (1.4) and (1.13). Let us consider now the case 1 p 2. Let z = u 1 - u2. 249

By the monotonicity condition (1.6) and by Holder’s inequality we have

Since 2(P- 1)(2-p)lp 2, we obtain

By (1.14) and by Holder’s inequality we have

Therefore

The conclusion follows now from (2.7) and (2.8). 250

The following theorem shows that the continuous dependence on f of the solutions u of (2.2) is uniform with respect to JL.

THEOREM 2.3. Let it E let E H1,P(Q) n let fl, f2 E H-1’q(S2), and let u I, U2 be the solutions of (2.2) corresponding to f = ,fl and f = f2. If 2 p then

where C is a constant depending only on p and co. If 1 p 2, then

where

and C is a constant depending only on p, co, cl.

PROOF. If we use v = u 1 - U2 as test function in the problems solved by by u I and U2, and if we subtract the corresponding equalities, we obtain

If 2 p +00, from (2.5) and (2.11) we obtain

which implies (2.9) by Young’s inequality. Let us consider now the case 1 p 2. From (2.6) and (2.11) we obtain

where the constant is given by Lemma 2.2. By (2.3) we have K(ul, u2, C r ( fl , f2, 1/1), and thus (2.10) follows from (2.12) by Cauchy’s inequality. 0 251

A connection between classical Dirichlet problems on open subsets of S2 and relaxed Dirichlet problems of the form (2.1 ) is given by the following remark.

REMARK 2.4. If U is an open subset of ~2, and v is a function of such that v = 0 q.e. in S2BU, then the restriction of v to U belongs to (see [3], Theorem 4, and [29], Lemma 4). Conversely, if we extend a function v E Ho’p (U) by setting v = 0 in QBU, then v is quasi continuous and belongs to Ho’p (S2) . Therefore, is the measure introduced in ( 1.1 ), then a function u in is the solution of problem (2.1 ) if and only if the restriction of u to U is the solution in of the classical boundary value problem

’ and, in addition, u = 0 q.e. in D Comparison principles. The solutions of relaxed Dirichlet problems satisfy the comparison principles given by the following propositions. PROPOSITION 2.5. e let f e H-1’q (S2), and let u be the solution of problem (2.1). If f > 0 in Q, then u > 0 q. e. in Q. PROOF. Let v = -(u A 0). Then v E n L~ (S2) 0 q.e. in Q. Since 0 q.e. in Q and f, v ) > 0, taking v as test function in (2.1) we obtain (Au, v) > 0. Since Dv = -Du a.e. in {u 0} and Dv = 0 a.e. in 0}, by the coerciveness condition (1.10) we have

As (Au, v) > 0, we obtain that v = 0 q.e. in Q, hence u > 0 q.e. in Q. 0 PROPOSITION 2.6. Let fl, f2 be Radon measures of (Q), let ILl, E and let u 1, u2 be the solutions of problem (2.1 ) corresponding to f l, itI and f2, Assume that 0 f2, f, f2, and A2 IL in Q. Then ul1 U2 q.e. ’ in Q.

PROOF. By Proposition 2.5 we have that U2 > 0 q.e. in Q. Let v - (Ul - Since 0 v and it2 ttl, we have v E u2)+. ui Lp1 (Q) C LpI-2 (Q). As taking v as test function in the problems solved by ul and U2 and subtracting the corresponding equalities, we obtain

Since

From the monotonicity conditions (1.4) and (1.6) it follows that v = 0 q.e. in Q and, consequently, u2 q.e. in Q. D 252

PROPOSITION 2.7. Let f l, f2 be Radon measures of H-l,q(0.), let i,c 1, i,c2 E Mb(0.), and let u 1, U2 be the solutions of problem (2.1) corresponding to fl, JLlI and f2, in 0., then lull u2 q. e. in 0.. PROOF. By Proposition 2.6 we have ul U2 q.e. in SZ. By (1.8) the function -u1 is the solution of (2.1 ) corresponding to - fi and it,, and so, by Proposition 2.6, we have also -u1 U2 q.e. in S2. 0 Estimates involving auxiliary Radon measures. We consider now some further estimates for the gradients of the solutions u of (2.2) and for the corresponding fields a (x, Du). We begin by proving that, if f E Lq(0.), then the solutions of (2.2) are actually solutions, in the sense of distributions, of a new equation involving a Radon measure À, which depends on u, it, f, and whose variation on compact sets can be estimated in terms of and

PROPOSITION 2.8. Let JL E let f E Lq (S2), and let u be the solution ofproblem (2.2) for some E H 1, p (o) f1 LP (S2). Letk, ~,2 be the elements of H - l,q (S2) defined by Au -f- ~, = f, A(u+) = f +, A(-u-) - ~,2 = - f -. Then À, X2 are Radon measures, ~,1 > 0, ~,2 > 0, À ),1 - ~.2, and IÀI ~.1 -~ ~2. Moreover for every compact set K 5; S2 we have

where ci is the constant in ( 1.11 ) and cp, Q is a constant depending only on p and Q.

PROOF. Let v E with v > 0 q.e. in Q and let Vn = Then Vn > 0 q.e. in Q and Vn E n L~(S2). As uvn > 0 q.e. in Q and f vn - f + vn a.e. in Q, by taking Vn as test function in problem (2.2) we obtain Since a.e. in ~Au, vn ) Jg,f+vndx::s DVn = n Dv {v nu+l and Dvn = Du+ a.e. in nu+l, we have

Taking the limit as n -~ oo, we obtain

Since Du+ = Du a.e. in fu+ > 0} and Du+ = 0 a.e. in lu+ = 0}, from (1.9) it follows that ~ ~ 253 for every v E Ho’p(~2) with v > 0 q.e. in S2. This implies that 0, hence ~.1I is a Radon measure. In a similar way we deduce that h2 is a non-negative Radon measure. By (1.9) we have Au = A(u+) + A(-u-), hence X = À2, which implies 1), ~.1 + À2. By the upper bound (I. 11) we have

where C depends only on p and Q. The same estimate holds for h2. To prove (2.13), for every 8 > 0 we fix a function z E such that z a 0 q.e. in Q, z > 1 q.e. in a neighbourhood of K, C p ( K ) -- . H6 (Q) Then

Taking the limit as E -~ 0 we obtain (2.13). C7 REMARK 2.9. Under the assumptions of Proposition 2.8, if f > 0 then u = u+ (Proposition 2.5) and X = Therefore in this case h a 0 and hence f in S2 in the sense of 0 The following theorem, together with Proposition 2.8, will be used in the proof of the main result of this section (Theorem 2.11): if un are the solutions of (2.2) corresponding to some measures pn E M§(Q), and (un) converges weakly in then (un ) converges strongly in for every r p.

THEOREM 2.10. Let (gn) be a sequence in (Q), let (Àn) be a sequence of Radon measures, and, for every n E N, let Un E be a solution of the equation

Assume that (un ) converges weakly in (Q) to some function u, (gn ) converges strongly in and (Àn) is bounded in the space of Radon measures, i.e., for every compact set K C S2 there exists a constant CK such that

Then (un) converges to u strongly in for every r p, and (a (x, DUn)) converges to a (x, Du) weakly in Lq (0, Re) and strongly in Ls (0, every s q. 254

PROOF. By Theorem 2.1 of [4] the sequence (un) converges to u strongly in for every r p. Let us fix a subsequence, still denoted by (un), such that (Dun) converges to Du pointwise a.e. in Q. As a (x, .) is continuous, we deduce that (a (x, Dun)) converges to a (x, Du) pointwise a.e. in Q. By the upper bound ( 1.11 ) this sequence is bounded in Lq (Q, and so, by the dominated convergence theorem, (a (x, DUn)) converges to a (x, Du) weakly in RV) and strongly in RV) for every s q. 0 As a consequence of Proposition 2.8 and Theorem 2.10 we have the following result. THEOREM 2.11. Let (gn) be a sequence in H - l,q (S2) which converges strongly to some g E H-l,q (Q), let be a sequence in MÖ (Q), and let (~n) be a sequence in such that JQ M for a suitable constant M independent of n. Assume that the solution Un of (2.2) corresponding to it = f = gn, 1/1 = 1/In converges weakly in to some function u. Then (un ) converges to u strongly in every r p, and (a (x, Dun)) converges to a (x, Du) weakly in Lq (Q, and strongly in MV) for every s q. PROOF. Given 8 E ]0, 1 [, we fix a function h E such that II h - E and we consider the solutions Zn of (2.2) corresponding to It = An, f = h, 1/1 = 1/In. By Theorem 2.3 we have

where a = l/(p 2013 1) for p > 2 and a = 1 for 1 p 2, while C is a constant depending only on p, co, cl, M, supn This implies, in particular, that (zn ) is bounded in Therefore, passing to a subsequence, we may assume that (zn ) converges weakly in to some function z, and (2.14) gives

By Proposition 2.8 there exists a sequence of Radon measures such that Azn + hn = h in Q. By (2.13) for every compact set K c Q the sequence (IÂnl(K)) is bounded. Therefore Theorem 2.10 implies that converges to z strongly in for every r p. Using Poincar6’s and Holder’s inequalities we obtain which, together with (2.14) and (2.15), gives

As E is arbitrary, we obtain that (un) converges to u strongly in The convergence of (a (x, to a (x, Du) can be proved as in Theorem 2.10. 0 255

3. - Corrector result

Definition of the corrector. Let (An) be a sequence of measures of and let f E L°° (S2). For every n E N let us consider the solution un of the problem

By estimate (2.3) the sequence (un) is bounded in H6’ (Q), thus we may assume that (un ) converges weakly in to some function u. By Theorem 2.11 the sequence converges to D u weakly in and strongly in rr) for every r p. We shall improve this convergence of the gradients and we shall obtain their strong convergence in LP(Q, rr) by means of a corrector: we shall define a sequence of Borel functions depending on the sequence (ILn), but independent of f, u, un, such that, if Rn is defined by

then the sequence (Rn ) converges to 0 strongly in Lp (S2 , This fact means that the oscillations around Du of the sequence of the gradients (Dun) are determined, up to a remainder which tends to 0 strongly in only by the values of the limit function u and by the sequence of correctors which depends only on the sequence (it,). In order to construct Pn, let us consider for every n E N the solution wn of the problem

By estimate (2.3) the sequence (wn) is bounded in thus we may assume that (wn) converges weakly in to some function w. Then we define the Borel function Pn : S2 ~ by

We are now in a position to state the main theorem of this section. 256

THEOREM 3.1. Let sequence of measures and let f e For every n E N, let un and wn be the solutions ofproblems (3.1 ) and (3.3). Assume that (un ) and (wn ) converge weakly in some functions u and w, and define Pn and Rn by (3.4) and (3.2). Then (Rn ) converges to 0 strongly in

REMARK 3.2. Let wo be the unique function of such that A wo = 1 in Q. By the comparison principle (Proposition 2.7) and by the homogeneity condition (1.3) we have c wo q.e. in ~2, with c = hence q.e. in Q. As (see [34], Chapter 4, Theorem 7.1), the functions u and w belong to and the sequences (un) and ( wn ) are bounded in D REMARK 3.3. Before proving Theorem 3.1, let us observe that, if f E then the sequence (Rn) defined by (3.2) and (3.4) converges to 0 weakly in and strongly in for every r p. Indeed § E > 0 } ) by Remark 3.2 and

while ( D un - Du) and ( D wn - Dw) converge to 0 weakly in and strongly in for every r p by the definition of u and wand by Theorem 2.11. The result of Theorem 3.1 is thus to assert that the convergence of ( Rn ) is actually strong in The corrector result of Theorem 3.1 is formally equivalent to the strong convergence of (un - to 0 in This assertion, which is only formal since w can vanish on a set of positive Lebesgue measure, becomes correct in if U is some open subset of [2 such that w > s > 0 in U (see Lemma 3.4 and (3.7)). 0 Three lemmas. To prove Theorem 3.1 we use the following lemmas. LEMMA 3.4. Assume that the hypotheses of Theorem 3.1 are satisfied. For every 6’ > 0 { w > s) f1 ( > Then for every B > 0 thefunctions ~~ belong to ([2) n ([2) and one has

Note that = w in Us and thus (3.6) implies formally that (Dun - D( QQ)) converges to 0 strongly in As D(uwn) is not always well defined, the rigorous formulation of the previous statement is that (3.6) yields 257 strongly in ,1I~’ ) . Since (~) is bounded in (Remark 3.2) and (up to a subsequence) converges to 1 almost everywhere in Us, while -!!- E (Remark 3.2), we conclude that for E > 0 fixed (3.6) implies

strongly in PROOF OF LEMMA 3.4. Define for every E > 0 and for every n E N the functions

First step. In this step we shall prove that these functions belong to L°° (S2) f1 and study their convergence as n - 00 for s > 0 fixed. The functions u and w~~ belong to while the sequences (un ) and (wn ) are bounded in (Remark 3.2) and converge to u and w weakly in Since any function in n belongs to for any JL e wv belongs to Thus ug and rn belong to f1 L°° (S2) f1 and and (r~) are bounded in and converge to w~~ and u - w~£ weakly in Ho’ p (S2) . By Theorem 2.11 the sequences (un) and (wn) converge to u and w strongly in for every r p, and so converges to wvE strongly in Therefore, passing to a subsequence, we may assume that (un), (wn), (Dun), (Dwn), converge to u, w, Du, Dw, D( wv£ ) almost everywhere in Q. Then the upper bound ( 1.11 ) and the continuity of a (x,.) imply that ~a (x , converge to a (x, Du), a (x, Dw), weakly in and almost everywhere in Q. As u - ~~ = 0 a.e. in Us, we obtain that (r~) converges to 0 strongly in (Dr~) converges to 0 weakly in and converges to a (x , D u ) weakly in

Let us fix a function w e Ho’p (S2) n L°° (S2) such that 0 ~ ~ ~ 1 q.e. in Q, w = 1 q.e. in U2e, w = 0 q.e. in and hence D~p = 0 a.e. in For instance, we can take cp = ~E (w) ~E ( w~£ ), where 4$s: R - R is the Lipschitz function defined by - 0 e, = ~ - 1 for e t 2e, = 1 for t > 2s. By the previous remarks the sequence converges to 0 weakly in and strongly in LP(Q).

Second step. Let us define 258

In this step we shall prove that for E > 0 fixed we have

For that we write En as

Since w = 0 q.e. in the function z = belongs to L°°(S2). This is trivial if p > 2. When 1 p 2 it is enough to observe that z = ~£ ( w~~ ) ~p, where BI1e is the locally Lipschitz function defined by (4.3) below. By the homogeneity property (1.3) we obtain

As = = in we wn wVSu U1,n’ taking v WVE wvernq;wVs nu e (3.3) get 259

and so, taking v = r§w in (3 .1 ), from (3.9) we obtain

Since E L°° (UE), and (r§) is bounded in L°° (S2) and converges to 0 strongly in LP(Us), while the sequences (a (x, DuE)) converge weakly in it follows that the sequences (,~l ), (jn2), (,~3 ), (:1;) tend to 0. To conclude the proof of (3.8) it is enough to show that

Since (Dwn) and (Du’) converge to Dw and Du almost everywhere in U,, it follows that

Let us prove that is equi-integrable. Let us consider the case 2 p Since u E n L°°(S2) and (wn) is bounded in L~(~), by the continuity condition (1.5) there exists a constant C such that 260

By Holder’s inequality for every measurable set E c S2 we have

By (3.12) this inequality shows that is equi- integrable. In the case 1 p 2, by the continuity condition (1.7) we have

and so the sequence ’a (x, (x, is dominated by an integrable function, since and is bounded in Therefore, in both cases I a (x, u (x, Du-) Iq is equi-integrable, and thus, by the dominated convergence theorem, (3.11) implies that (a (x, £ Dwn ) - a (x, Dun) ) converges to a (x, (x, Du) strongly in As converges to 0 weakly in LP(Q, kv), we obtain (3.10), which implies (3.8).

Third step. If 2 p then (2.5) gives

If 1 p 2, we note that the and are sequences (lIun IILPIn g) n ) bounded by estimate (2.3). Since u and wE belong to Ho’p (S2) n L°° (S2), we conclude that is bounded too. Since (un ) and are bounded in by (2.6) there exists a constant K such that

Taking (3.13) and (3.14) into account, in both cases we obtain from (3.8) that

hence 261

As w v 2E = w V E q.e. in U2,, we have un - q.e. in U2, and a.e. in Therefore with 2E Drn - Dun - D( wv2E’ U2£’ (3.15) implies (3.6) replaced by s. 0

LEMMA 3.5. Let f E L°° (S2) and, for every n E N, let Un be the solution of (3.1 ). For every s > 0 define V, = {w e }. Then

PROOF. For every 8 > 0 let 1>£: ~ -* ~ be the Lipschitz function defined at the end of the first step of the proof of Lemma 3.4, and let z £ E n L 00 (Q) be the function defined by z£ = 1 - ~~ (w). As z’ > 0 q.e. in Q and z£ = 1 q.e. in V£, by (1.10) and (3.1) we have

Since (un ) converges to u strongly in LP(Q) and is bounded in (Re- mark 3.2), while converges to a (x, Du) weakly in (Theorem 2.11), we can take the limit of the the last two terms as n - oo, obtaining

As (ze) is bounded in L’ (0) and converges to the characteristic function of f w = 0} as 8 - 0, while u = 0 a.e. in tw = 0} (Remark 3.2), we have that (uz’) converges to 0 strongly in LP(Q). On the other hand, since c w q.e. in Q by Remark 3.2, we have

so that converges to 0 strongly in LP( , Taking the limit in (3.17) as E --~ 0 we obtain (3.16). 0 262

LEMMA 3.6. Let f E L°° (S2) and, for every n E N, let Un be the solution of (3 .1 ). For every E > 0 define W, = { w Ewl. Then

PROOF. For every E > 0 let 1>£: 1R -* R be the Lipschitz function defined at the end of the first step of the proof of Lemma 3.4. As £ E (Remark 3.2), the function z’ = 1- I>£(~~£) belongs to f1 L°° (S2). As 0 q.e. in S2 and z’ = 1 q.e. in We, by (1.10) and (3.1) we have

Since (a (x, converges to a (x, Du) weakly in (Theorem 2.11) and (un ) is bounded in L°°(Q) and converges to u strongly in LP(Q) (Re- mark 3.2), we can take the limit of the the last two terms as n --~ oo, obtaining

As (z’) is bounded in and converges to the characteristic function of f u = 01, we have that (uz’) converges to 0 strongly in LP(Q). Moreover,

and so (uDze) converges to 0 strongly in as 6’ --~ 0. Therefore (3.18) follows from (3.19) by taking the limit E -~ 0. 0 263

PROOF OF THEOREM 3.1. Recall that Us = { w ( and w, = { w n w }, which for s > 0 allows us to write

Since Rn = Dun - Du - Dw) in lw > 0}, we deduce from (3.7) that for 8 > 0 fixed -

On the other hand we shall prove that

Since ~I c w q.e. in S2 (Remark 3.2), we deduce from (3.5) that Dul + ] q.e. in S2. Therefore

for every E > 0. c w (Remark 3.2), we have Du = Dw = 0 a.e. in f w = 01. This fact, together with Lemma 3.5 (applied to the sequences (un) and (wn)), allows us to obtain (3.20) from the previous inequality. E w q.e. in W,, by (3.5) we have Dun - Dul + B D wn - D w ~I q. e. in W~ . Therefore

As the characteristic function of W£ converges to the characteristic function of tw > 0} n {u = 0}, and Du = 0 a.e. in ju = 0}, the previous inequality together with Lemma 3.6 yields (3.21), which concludes the proof of the theorem. 0 Theorem 3.1 explains how to correct Dun in order to obtain strong convergence (see (3.2)). Note that the function Du -+ u Pn is not a gradient in general. The following theorem gives a corrector result in for the function un. 264

’ THEOREM 3.7. Let a sequence ofmeasures of Mg (Q) and let f E Assume that the solutions Un and w,~ of problems (3.1 ) and (3.3) converge weakly in Ho’p (S2) to some functions u and w. Then for every E > 0 we have

with lim lim sup = 0. n-+oo 0 ’"~ PROOF. Recall that Ue = { w > ~ } n I > { w ~ }, and We = {w > 81 n E wl. Then

Since, by Lemma 3.4, (Dr~) converges to 0 strongly in as n 2013~ oo, we have only to estimate the last two terms of (3.22). As

and lul ::S c w q.e. in S2 (Remark 3.2), we have

Since (wn ) is bounded in L°°(Q) (Remark 3.2) and converges to w weakly in 7-fo’~(~), we obtain

As c w q.e. in S2, we have Du = 0 a.e. in { w = 0}, and so the last two terms tend to 0 as E --~ 0. Therefore, Lemma 3.5 (applied to the sequences (un) and (wn)) implies that

Since q.e. in W,, we have 265 and thus

As the characteristic function of W, converges to the characteristic function of {w > 0} n ju = 0}, and Du = 0 a.e. in ju = 0}, the term fWe IDulPdx tends to 0 as E ~ 0. Therefore, by Lemma 3.6 we have

which concludes the proof of the theorem. 1:1 The case f ~ requires a further approximation (see [11]) and will be considered at the end of Section 6.

4. - Estimates based on the corrector result

Let a sequence of measures of and let f E Assume that the solutions un and wn of problems (3.1) and (3.3) converge weakly in to some functions u and w. In this section we shall study the behaviour of the sequences

where /3 > (p - 1) v 1 and ~ E n As explained in the introduction, these estimates will be useful in the proof of the main results of Section 6. Since for 1 p 2 the function £ does not belong to H6’ I p (Q) (note that the derivative of the function t H ~ singular at the origin), formula (4.1) is not correct and we are forced to introduce the locally Lipschitz function R defined by

and to replace by in (4.1) and (4.2). This leads to some technical estimates that will be used in Section 6. We begin with an estimate in the set { w > ~ } n I > 6’ w 1. 266

LEMMA 4.1. Let sequence of measures and let f E Assume that the solutions un and wn of problems (3.1 ) and (3.3) converge weakly in some functions u and w. > 0, let /3 > 1, and let Ve E f1 the function defined by Ve = where We is given by (4.3). Then the sequence

converges weakly in Ll 1 (Ue), as n - 00, to the function

PROOF. By Theorem 3.1 we have

where ( Rn ) converges to 0 strongly in Since v, = a.e. in U,, by the homogeneity property (1.3) for a.e. x E U, we have (4.5)

Similarly, we obtain

By Theorem 2.11 the sequences (un ) and (wn ) converge to u and w strongly in H6,r (Q) for every r p, and so, passing to a subsequence, we may assume that (un), (wn), (Dun), (Dwn) converge to u, w, Du, Dw almost everywhere in As a (x,.) is continuous for a.e. x E S2, this implies that (a (x, and (a (x, converge to a (x, Du) and a (x, -~-Dw) almost everywhere in Us . We want to prove that (a (x, (x, converges to a (x, Du) - a (x, strongly in To this aim it is enough to show that the sequence I a (x, Dun) - a (x, is equi-integrable. Let us consider first 267 the case 2 p +oo. Since " E (Remark 3.2), by (1.5) and (4.4) there exists a constant C such that

for a.e. x E Us. By Holder’s inequality for every measurable set E c U, we have

As is bounded in and ( Rn ) converges to 0 strongly in LP(r2, MV), the previous inequality, together with (4.7), shows that I a (x, Dun)- a (x, is equi-integrable in Ue. In the case 1 p 2, by the continuity condition (1.7) we have

As (Rn) converges to 0 strongly in LP(Q, the sequence l a (x, Dun) - is equi-integrable in Us. Therefore, in both cases the dominated convergence theorem implies that (a (x, converges to a (x, Du) -a (x, w D w) strongly in rr). As (Dwn) converges to Dw weakly in and is bounded in (Remark 3.2) and converges to w almost everywhere in Q, it follows that

converges to

weakly in As ( a (x , D wn ) ) converges to a (x , D w ) weakly in (Theorem 2.11), the sequence (a (x, wf) converges to (a (x , D w ) , DVe)wf3 weakly in The conclusion follows now from (4.5) and (4.6). 0 268

LEMMA 4.2. Let a sequence of measures of A4 0 (Q) and let f E L°°(S2). Assume that the solutions Un and Wn of problems (3.1 ) and (3.3) converge weakly in Ha’p (Q) to some functions u and w. Let /3 > 1, let cP E H6" P (Q) f1 L 00 (Q), and, for every E > 0, let V, E n L°° (S2) be the function defined by v, = where We is given by (4.3). Then

with lim lim sup I = 0. o n_m PROOF. For every c > 0 we define !7p = { w > } n El, and We = lw > El n w}. Then

where

Similarly, we define Be, C’ by replacing un with u and wn with w, so that

By Lemma 4.1 we have

Since (a (x , D u n ) ) and (a (x , D wn ) ~ converge to a (x , D u ) and a (x , D w ) weakly in (Theorem 2.11), and (wn) is bounded in LOO(Q) (Re- mark 3.2) and converges to w strongly in while Ve E (Re- mark 3.2), we conclude that (4.9) limC-C’

Let us consider now the term For every measurable set B we define 269

Similarly, we define :11 (B), ,7£~2(B), :1e,3(B) by replacing un with u and wn with w. Clearly we have

Since /3 > 1, the sequence is bounded in (Remark 3.2). More- 1 over, as lul c w (Remark 3.2), by (4.3) we have q.e. in S2 . Therefore, by the upper bound (1.11) there exists a constant K such that for s s 1 we have

and thus Lemmas 3.5 and 3.6 imply that

A similar estimate shows that

1 Since q.e. in W,, by (4.3) we have £p-1 q.e. in W~ . Therefore, the boundedness of in (Remark 3.2) and the upper bound (1.11) imply that

for a suitable constant K. As (wn) is bounded in we conclude that

A similar estimate shows that

Since (a (x, Dwn)) converges to a (x, Dw) weakly in Lq (Q, MV), and (wn) is bounded in (Remark 3.2) and converges to w strongly in LP(Q), we conclude that

From (4.10)-(4.15) we obtain

As 7Zn = .An - ,,4~ the conclusion follows from (4.8), (4.9), (4.16). D 270

LEMMA 4.3. Let a sequence of measures and let f E L°°(S2). Assume that the solutions Un and Wn of problems (3.1) and (3.3) converge weakly in Ho’p (SZ) to some functions u and w. Let 8 > 0, let fl > ( p - 1 ) v 1, and let as in the 3.4. Then U8n - uwnwvE proof of Lemma

tends to 0 as n -~ oo for every ~p E n L°° (S2).

PROOF. Let w E Ho ’ p ( S2 ) n L °° ( S2 ) and let rn = u n - u n as in the proof of Lemma 3.4. Since the sequences (un) and are bounded in (Re- mark 3.2), by ( 1.13) and ( 1.14) there exists a constant C such that Moreover, since f3 (p - 1) v 1 and (wn) is bounded in (Remark 3.2), there exists a constant K such that Therefore

The conclusion follows now from estimate (2.3) and from Lemma 3.4. 0

LEMMA 4.4. Let be a sequence of measures let f E L°° (S2), and let cP E rl L °° ( S2). Assume that the solutions un and Wn ofproblems (3 .1 ) and (3.3) converge weakly in Ho’p (S2) to some functions u and w. For every 8 > 0 let V, E n L ’ (Q) be the function defined by V£ = ~£ ( w~£ ), where T, is given by (4.3). Let /3 > ( p - 1 ) v 1 and let

Then lim limsup I = 0. 0

PROOF. For every E > 0 we define { w > s) n > u}, Ve = w{ - eI , and W£W ==_ {w{ > Is) n{I {lul I ] ::S- e w . As = u£n q.e.’’ in U, by Lemma 4.3 for every B > 0 the sequence 271 tends to 0 as n - oo. As cp is bounded, to conclude the proof it is enough to show that

Since /3 > 1, we have for a suitable constant K (Remark 3.2). Therefore

and thus (4.17) follows from (2.3) and from Lemmas 3.5 and 3.6. Moreover, as 1 lul ~ c w (Remark 3.2), by (4.3) we have q. e. in S2 . Therefore

and so (4.18) follows from Lemma 3.5. Since s w q.e. in W,, by (4.3) 1 we have IVel q.e. in W,. Therefore

and thus (4.19) follows from (2.3). 0

5. - Constructing the measure from the solution corresponding to f = 1

In this section we shall study the properties of the set of the functions w such that

q.e. in S2 , and Aw 1 in D’(Q).

. The results obtained in the present section will be crucial in the proof of Theorems 6.3 and 6.5. They do not depend on the results of Sections 3 and 4, but only on those of Sections 1 and 2. 272

By the coerciveness condition (1.10) for every w E IC(Q) we have

This shows that is bounded and hence weakly relatively compact in 7~0 ~(~). Let wo be the solution of the Dirichlet problem

By the comparison principle (Proposition 2.6) we have 0 w wo q.e. in S2 for every w E As Wo E (see [34], Chapter 4, Theorem 7.1), the set lC(Q) is also bounded in L°°(Q). Given w E we define

By the definition of J’C(S2) we have v > 0 in D’(0), hence v is a non-negative Radon measure. As Aw E we have v E Our aim in this section is to prove the following theorem, which characterizes as the set of the solutions of all relaxed Dirichlet problem corresponding to f = 1. THEOREM 5. l. The set is compact in the weak topology of H6’ p (Q). A function W E H6’ (Q) belongs to J’C(S2) if and only if there exists it E (Q) such ° that w is the solution of the problem

The measure it E MÖ (Q) is uniquely determined by W E More precisely, for every W E J’C(S2) and for every Borel set B C Q we have

where v is the non-negative measure of H-l,q(r2) defined by v = 1 - Aw. Note that in view of (5.2) we have

for every Borel set B c S2 . To prove the theorem we need some technical results. We begin with a penalization lemma. 273

LEMMA 5.2. Let p E and let u E f1 L~ (S2). For every n E N let un E n the solution of the problem

Then (un ) converges to u strongly in Ho’p (S2) and in L~ (Q). PROOF. Taking v = un - u as test function in (5.4) we obtain

hence

If 2 p +00, from (2.5) we get

hence

By using Young’s inequality we obtain 274

This shows that (un ) converges to u weakly in and in L§(Q), and by (5.6) this implies also that (un) converges to u strongly in and in L~(S2). Assume now that 1 p 2. If we apply (2.8) to the Lebesgue measure we obtain

This inequality, together with (2.6) and (5.5), gives

where

By using Young’s inequality we obtain

where and K2(u) are constants depending on u. This implies that

and shows that (un) converges to u weakly in H6’ (Q) and in Therefore (H(un, u)) is bounded and by (5.7) (un) converges to u strongly in and in L~ (S2) . 0 275

Condition (ii) in the definition of Mg (Q) (see Remark 1.2) plays a crucial role in the following lemma, that will be used in the proof of the uniqueness result given by Lemma 5.4. LEMMA 5.3. Let E and let w be the solution of problem (5.1). Then = +oo for every Borel with Cp(B f1 {w = 0}) > 0.

PROOF. Let u E with 0 ::S u 1 q.e. in Q, and, for every n E N, let u n E be the solution of problem (5.4). By the comparison principles (Propositions 2.5 and 2.6) and by the homogeneity condition (1.3) we have 0 ::S ni/(p-1)w q.e. in Q, hence un - 0 q.e. in {w = 0}. Since by Lemma 5.2 (un) converges to u in H1’p(S2), we have u = 0 q.e. in {w = 0}. Let U be a quasi open subset of S2 such that +oo. By Lemma 1.1 there exists an increasing sequence (Zn) in converging to 1u quasi everywhere in Q and such that 0 ::S 1u q.e. in S2 for every n E N. As +oo, each function zn belongs to hence zn = 0 q.e. on {w = 0} by the previous step. This implies that Cp (U n {w = 0}) = 0. Let us consider a Borel set B with Cp(B f1 {w = 0}) > 0. For every quasi open set U containing B we have Cp(U n { w = 0} ) > 0, hence = +oo by the previous step of the proof, and so ~(B) _ -1-0o by property (ii) in the definition of 0 We are now in a position to prove the following uniqueness result. LEMMA 5.4. and i,c be measures Assume that there exists a function w E fl n L~ (S2) such that

Then À = It. PROOF. From the comparison principle (Proposition 2.5) we know that w > 0 q.e. in Q. Let us consider the measures ho and Ao defined for every Borel set B C Q

by - -

We want to prove that Ào = po. For every E > 0 let k, and it, be the measures defined by

To prove that ho = Mo it is enough to show that X, = ps for every 8 > 0. Let us fix 6- > 0. As w E n L~ (S2), hs and are bounded measures. 276

Therefore it is enough to show that hs (U) = for every open set U c SZ . Let us fix U and let Us = U fl { w > ~ }. As Us is quasi open, by Lemma 1.1 there exists an increasing sequence (zn ) in converging to quasi everywhere in S2 and such that 0 q.e. in S2 for every n E N. As w E n and w > 8 q.e. in Us, we have k(U,) +oo and p(Us) +oo, hence zn E n for every n E N. From (5.8) and (5.9) we get

Taking the limit as n - oo we obtain

This shows that hs = for every 8 > 0, hence ÅO = /-to. For every Borel set B contained in f w > 0} we have

If B is Borel set contained in [w = 0} and Cp(B) > 0, then h(B) = JL(B) = + oo by Lemma 5.3. If C p ( B ) = 0, then k (B) = A (B) = 0 by the definition of Therefore X(B) = h(B n {w > 0}) + h(B n lw = 0}) - JL(B n tw > 0}) + p (B n tw = ol) = it(B) for every Borel set B c S2. 0

PROOF OF THEOREM 5.1. Let us prove that is weakly compact in Let (wn) be a sequence in As IC(Q) is bounded in we may assume that ( wn ) converges weakly in to a function w, and we have only to prove that w E As wn a 0 q.e. in Q, we have also w > 0 q.e. in Q. To prove that A w 1 in Q, we consider the measures vn = 1 - A wn . By the definition of (vn ) is a sequence of non-negative Radon measures which belong to Since (wn) is bounded in the upper bound (1.11) implies that (vn ) is bounded in and thus, since vn is non-negative, the sequence is bounded for every compact set K C Q. By Theorem 2.10 the sequence (a (x, converges to a (x, Dw) weakly in e), and hence (Awn) converges to Aw weakly in As 1 in D’(Q), we conclude that 1 in D’(Q), hence w E In the rest of the proof we follow the lines of Theorem 1 of [13] and of Proposition 3.4 of [21]. Let it E A4j(Q) and let w be a solution of (5.1). Then w > 0 q.e. in Q by the comparison principle (Proposition 2.5). From Proposition 2.8 and Remark 2.9 we deduce that 1 in D’ (S2), and we conclude that w E

Conversely, assume that w E Let v = 1 - A w and let JL be the measure defined by (5.2). We first prove that it E MP(Q). Since the measure v 277 is non-negative and belongs to we have v(B) = 0 and thus = 0 for every Borel set with Cp(B) = 0. It remains to prove that

(5.10) = quasi open , B c U } for every Borel set B c S2 with +oo. For every n E N let J1-n be the measure on S2 defined by J1-n(B) = p(B n { w > ~}). Note that = > n }) np-1 v(fw > n }) np fs2 w dv = Aw, w) +oo. Let us fix a Borel set B c S2 with +oo. By the definition of J1- we have {w = 0}) = 0. For every n > 2 let Bn = B n {~ ~ ~ ~y}, and let w}, so that Since +00, for every s > 0 and for every n E N there exists an open set Vn, with Q, such that + £2-n = A(Bn) + ~2~. Let Un = Vn n lw > ~}. As w is quasi continuous, the set Un is quasi open. Moreover Bn C Un and lt(Un) = J1-n (Vn) A(Bn) + ~2’". Let Uo - B n tw= 0 } and let U be the union of all sets Un for n > 0. Then U is quasi open, contains B, and + s. Since E > 0 is arbitrary, this proves (5.10). Let us prove that w is a solution of (5.1 ). By (5.2) we have

hence w E L~ (S2). Let v E n L§(Q). By (5.2) we have v = 0 q.e. in [w = 0}. By the definitions of p and v we have

which proves (5.1 ). The uniqueness of JL follows from Lemma 5.4. 0 The following density result will be crucial in the proof of Theorem 6.3. PROPOSITION 5.5. Let JL E let w be the solution of problem (5.1), and let /3 :::: 1. Then the set w E dense in f1 L~ (S2).

PROOF. First we note that, since w E (Remark 3.2) and /3 > 1, the function wfJ cP belongs to Ho ’ p ( SZ ) n L ~ ( S2 ) for every w E To prove the density, for every u E we have to construct a sequence in such that converges to u both in and in L~ (S2). Since every function of can be approximated by truncation both in and in L~(~2), we may assume that u E Ho ’ p ( S2 ) n L °° ( S2 ) n L ~ ( S2 ) and that 0 q . e . in Q. For every n E N let un be the solution of (5.4). By the comparison principles (Propositions 2.5 278 and 2.6) and by the homogeneity condition (1.3) we have 0 c w q.e. in ~2, where cp-1 = By Lemma 5.2 the sequence (un ) converges to u both in and in Therefore, in our approximation problem it is not restrictive to assume also that there exists c > 0 such that 0 _ u cw q.e. in S2. Since {(u - cE)+ > 0} c fw > sl, and (u - cE)+ converges to u, as 6- - 0, both in and in LP (0), we may also assume that there exists E > 0 such that lu > 0} c {w > El. Then u/(w v E)P. As u E n L°°(Q), we have UIWP E n Therefore there exists a sequence (wn) in bounded in which converges to .z = ulwo strongly in and quasi everywhere in Q, hence in Q. Since w E n and /3 > 1, the sequence converges to u strongly in Ho’p (S2) . As w E n (Remark 3.2) and 1, the function wfJ belongs to f1 Since (CPn) is bounded in L°°(Q) and converges to z = ulwo ti-a.e. in Q, by the dominated convergence theorem the sequence converges to = u strongly in D

6. - y A -convergence: compactness and localization properties yA-convergence. In this section we introduce the notion of yA-convergence in A4p(Q), which is defined as the convergence of the solutions of the corresponding relaxed Dirichlet problems. When p = 2 and A is the Laplace operator -A, this notion is defined in [23] in terms of the r-convergence of the functionals In IDul2dx + In U2dit associated with the relaxed Dirichlet problems. For the extension of this definition to the case of symmetric linear operators we refer to [6] and [17]. For the case of non-symmetric linear operators see [21]. The case where A is the subdifferential of a convex functional In Du) dx defined on with 1 p +0oo, is studied in [20] by r-convergence techniques. The definition we introduce below involves only the solutions of the relaxed Dirichlet problems (2.1), and coincides with the previous definitions in all cases already considered in the literature. DEFINITION 6.1. Let be a sequence of measures of and let p E A4p(Q). We say that yA-converges to p (in Q) if for every f E the solution Un of the problem

converges weakly in Ho ’ p ( S2), oo, to the solution u of the problem 279

Let us emphasize the fact that the notion of y A -limit depends on the operator A. Although the definition depends also on Q, we shall see in Theorems 6.11 and 6.12 that the boundary condition on aQ does not play an important role in this problem. REMARK 6.2. Since the solutions of (6.1) depend continuously on f, uni- formly with respect to it (Theorem 2.3), a sequence y A -converges to J1 if and only if the solutions of (6.1 ) converge weakly in the solution of (6.2) for every f in a dense subset of 0 Let be a sequence of measures of Mg (Q) and let J1 E Let wn and w be the solutions of the problems

The following theorem characterizes the yA-convergence of a sequence of measures in terms of the weak convergence in of the sequence THEOREM 6.3. Let sequence of measures and G Let wn and w be the solutions of problems (6.3) and (6.4). The following conditions are equivalent: (a) (wn) converges to w weakly in (b) JL. PROOF. To prove that (b) implies (a) it is enough to take f = 1 in the definition of yA-convergence. Conversely, assume that (a) holds true. Given f E we consider the solutions un of problems (6.1). By estimate (2.3) the sequence (un) is bounded in so we may assume that (un) converges weakly in to some function u. We want to prove that u is the solution of (6.2). By the comparison principle (Proposition 2.7) and by the homogeneity condition (1.3) we have c wn q.e. in ~2, with c = On letting n - oo we get c w q.e. in Q. For every s > 0 let be the locally Lipschitz function defined by (4.3) and let Ve E be the function defined by Ve = Let us fix w E and f3 > (p - 1) v 1. As wn E n LOO(r2) (Remark 3.2), by taking v = in (6.1) and v = in (6.3) we obtain 280

By subtracting the second equation from the first one we get

By Lemmas 4.2 and 4.4 we have

with

Since (wn) is bounded in (Remark 3.2) and converges to w strongly in for every 8 > 0 we have

Together with (6.5) and (6.6) this implies that

with

Let v = By Theorem 5.1 w belongs to thus v is a non-negative Radon measure of H - l,q (Q). Therefore we have

1 c w q.e. in SZ (Remark 3.2), by (4.3) we have q.e. in Q. From definition of We we obtain that converges to quasi everywhere in Q (recall that 13 > p - 1). Since v E and w~ is bounded (Remark 3.2), we have wfJcp E and thus, by the dominated convergence theorem, 281

Therefore (6.7) implies

c w q.e. in SZ, and w E we have u E and, by (5.3),

and thus (6.8) gives

Since the set f w~~p : w E is dense in (Proposi- tion 5.5), it follows that u coincides with the solution of (6.2). Therefore (pn) yA-converges to JL by Remark 6.2. D REMARK 6.4. The uniqueness of the yA-limit is an easy consequence of Theorem 6.3. Indeed, if yA-converges to JL and À, then w satisfies (5.8) and (5.9), so that JL = À by Lemma 5.4. 0 Compactness and density properties. The following theorem proves the compactness of Mg (Q) with respect to yA-convergence. THEOREM 6.5. Every sequence of measures contains a yA-con- vergent subsequence. PROOF. Let be a sequence of measures of Mg (Q) and, for every n E N, let wn be the solution of problem (6.3). By Theorem 5.1 each function wn belongs to the set defined at the beginning of Section 5. Since is compact in the weak topology of a subsequence of (wn) converges weakly in to some function w E By Theorem 5.1 there exists a measure JL E Mg (Q) such that w is the solution of problem (6.4). The conclusion follows now from Theorem 6.3. 0

The case of Dirichlet problems in perforated domains is a particular case of the previous one. It is considered in the following theorem. THEOREM 6.6. Let (Qn) be an arbitrary sequence of open subsets Then there exist a subsequence, still denoted by (S2n ), and a measure JL E such that for every f E the solution un of the problem

extended by 0 in SZ B Qn, converges weakly in Ho’p (S2) to the solution u of problem (6.2). 282

PROOF. The conclusion follows easily from the compactness theorem (Theo- rem 6.5) and from the fact that each function un can be regarded as the solution of problem (6.1 ) with = (Remark 2.4). D Using Theorem 6.3 we can prove the following density result in .~lo (S2). We shall see in Corollary 6.10 that the strong convergence in of the sequence (wn) implies the strong convergence in of the sequence (un) of the solutions of (6.1) for every f E

PROPOSITION 6.7. Every E the of a sequence of Radon measures such that the solution wn of (6.3) converges strongly in the solution w of (6.4).

PROOF. By (5.2) a measure p E is a Radon measure if the solution w of (6.4) satisfies

(6.9) inf w > 0 for every compact set Q . K

Now let Ho ’ p ( S2 ) be the solution of the equation = 1 in 2. By the strong maximum principle (see [34]) wo satisfies (6.9). Let us fix E MÖ (Q) and let w E be the solution of (6.4). For every n E N, let us define It is easy to see that wn is a non-negative subsolution of the equation Au - 1, hence wn E (see, e.g., [30], Theorem 3.23). Moreover the functions wn satisfy (6.9) and converge to w strongly in Therefore the measures E associated with wn according to (5.2) are Radon measures and y-converge to p by Theorem 6.3. D

Strong convergence and correctors. The following theorem deals with the convergence of solutions, momenta, and energies, when also f varies.

THEOREM 6.8. Let sequence of measures which yA-converges to a measure ,c E and let ( fn) be a sequence in which converges strongly to some f E For every n e let un be the solution of the problem

and let u be the solution of problem (6.2). Then the sequence (un) converges to u weakly in and strongly in for every r p. Moreover (a (x, Dun)) converges to a (x, Du) weakly in Lq and strongly in LS (Q, for every s q. Finally the energy (a (x, Dun ) , Dun ~ + converges 283 to (a (x, Du), Du) + weakly* in the sense of Radon measures on Q, i. e.,

for every w e

PROOF. For every n e N, let vn be the solution of problem (6.1 ). By Theorem 2.3 the sequence (un - vn) converges to 0 strongly in By the definition of yA-convergence, ( vn ) converges to u weakly in Therefore (un) converges to u weakly in Moreover, by Theorem 2.11, (u n ) converges to u strongly in Ho ’r ( S2) for every r p, and converges to a (x , D u ) weakly in and strongly in for every s q. By (6.10) for every cp E we have

Since ( fn ) converges to f strongly in while (un ) converges to u weakly in and (a (x, DUn)) converges to a (x, Du) weakly in we conclude that

for every w E (in the second equality we used the fact that u is the solution of (6.2)). Since the Radon measures (a (x, Dun ) , Dun ) + are non-negative, an easy approximation argument shows that (6.11 ) holds for every 0

We consider now a corrector result for the strong convergence in Ho’p (Q). 284

THEOREM 6. 9. Under the assumptions of Theorem 6. 8 let ( Pn ) be the sequence of correctors defined by (3.4), where Wn and w are the solutions of (6.3) and (6.4). Then for every 8 > 0 there exists a function U’ E rl L°° (S2) n L~ (S2), with Ilu - ue II l,p E and I ueI c’ w for some constant c’ > 0, such that the H6’(0) (Q) - sequence (R’) defined by satisfies

If f E L°° (S2), we can 0 and u’ = u for every.F > 0. PROOF. When f belongs to L°°(Q) the corrector result is given by Theo- rem 3.1. When f belongs to for every 8 > 0, a > 0, K > 0 we can choose fe E such that If 2 p +00 we choose a = p - 1 and K = !C-’IP, where C is the constant which appears in (2.9). If 1 p 2, we choose a - 1 and K = M A 1, where M = + and C is the constant which appears in (2.10). We define vn as the solution of problem (6.1 ), vn as the solution of the analogous problem relative to fe, and u’ as the solution of problem (6.2) relative to fe. For 8 > 0 fixed the sequence (vn) converges to u’ weakly in Ho ’ p ( SZ ) . From the uniform continuity (Theorem 2.3) we deduce that

while Remark 3.2 implies that u’ belongs to and cew q.e. in S2, with c’ = The corrector result of Theorem 3.1 asserts that the sequence defined by

converges to 0 strongly in when E > 0 is fixed and n - oo. From (6.12) and (6.15) we obtain

We then deduce (6.13) from (6.14), (6.15) and from the fact that (un - vn) converges to 0 strongly in by Theorem 2.3. D COROLLARY 6.10. Under the assumptions of Theorem 6.8, if the solution wn of (6.3) converges strongly in the solution w of (6.4), then un converges to u strongly in 285

PROOF. For every B let u £ be the function introduced by Theorem 6.9. As ~ is bounded on {w > 0}, if (wn) converges to w strongly in then for B fixed (ue Pn) converges to 0 strongly in The conclusion follows now from (6.12) and (6.13). D

Localization properties. We conclude this section by showing the local character of the y-convergence. The following theorem deals with the convergence of local solutions in an open subset U of ~2, without any assumption on the boundary conditions satisfied on For every open set the duality pairing between N’~(~7) and is denoted by ~ ~ , ~ ) U . If we replace Q by U in ( 1.12), we obtain an operator from to which will still be denoted by A. THEOREM 6.11. Let be a sequence of measures which yA-converges in S2 to a E Let U be an open subset of Q, let ( fn) be a sequence in which converges strongly to some f E and let (un ) be a sequence in which converges weakly to some u E Suppose that

Then

Moreover, (un ) converges to u strongly in H l,r (U) for every r p, and (a (x, Dun ) ) converges to a (x, Du) weakly in Lq (U, rr) and strongly in rr) for every s q. Finally the energy (a (x, Dun ) , Dun ) + lunlpltn converges to (a(x, Du), D u ) + lu IP JL weakly* in the sense of Radon measures on U. PROOF. Let us fix an open set U’ CC U and a function ~ E C’ (U) such that ~ > 0 in U and ~ = 1 in U’. Using v = ~ un as test function in (6.16) we obtain

for a suitable constant M. By Theorem 2.11 the sequence (un) converges to u strongly in for every r p, while (a (x, converges to a (x, Du) weakly in and strongly in for every s q. Since (un) is bounded in and, by the upper bound ( 1.11 ), (a (x, is bounded in 286

Lq ( U, r’), from the arbitrariness of U’ C C U we conclude that (un ) converges to u strongly in for every r p, while converges to a (x, Du) weakly in L q ( U, r’) and strongly in for every s q. Let us fix U’ and ~ as before and let us consider the function cp defined by cp(x) = exp(l - 1/Ç(x)), > 0, and cp(x) = 0, if ~ (x ) = 0. Then cp E 0 in U, cp = 1 in U’, and E This implies, in particular, that E for every v E Let us define Zn = z = cpu, and

By the homogeneity condition (1.3) for every v E we have

Therefore (6.16) implies that zn is the solution of the problem

where gn is the element of defined by

for every v E Ho ’ p ( S2 ) . Let g be the element of defined by

for every v E 7:/o’~(~). Let us prove that (gn ) converges to g strongly in Since (a (x, Dun)) converges to a (x, Du) weakly in Lq (U, and E C’(U), the last two terms in the definition of gn converge strongly in to the corresponding terms in the definition of g. It remains to prove that (1/In) converges to 1/1 strongly in Since (un) converges to u strongly in for every r p, passing to a subsequence, we may assume that (un ) converges to u and (Dun ) converges to Du almost everywhere in U. Since a(x, .) is continuous, we have also that converges to 1/1 almost everywhere in U. If 2 p +oo, by (1.5) there exists a constant C such that 287 for a.e. x E U. As supp(q;), by Holder’s inequality for every measurable set E c U we have

where K = supp((p). Since (un) converges strongly in LP(K), the sequence is equi-integrable on U. If 1 p 2, by (1.7) we have and is equi-integrable on U in this case too. As converges to p almost everywhere in U, by the dominated convergence theorem (pn) converges to ljI strongly in Therefore (gn ) converges to g strongly in Since (zn) converges to z weakly in by (6.18) and by Theorem 6.8 the function z is the solution of the problem

As w - 1 in U’, we have u = z q.e. in U’, hence u E L~ (U’). Moreover, if v E with supp(v) CC U’, then (g, v) = ( f, v), and so (6.19) implies (6.17). The convergence of the energy can be proved as in Theorem 6.8. 0

THEOREM 6.12. Let (J1n) a sequence of measures which verges in S2 to a measure J1 E and let U be an open subset Then yA-converges to IL in U. PROOF. Let us fix f E For every n E N let u n be the solution of problem (6.1), with Q replaced by U. By estimate (2.3) we know that a subsequence, still denoted by (un ), converges weakly in Ho ’ p ( U) to a function u E Then Theorem 6.11 u E for set U’ C C U 7~((/). by ’ every open and u is a solution of problem (6.17). It remains to prove that u E L~((/). It is easy to construct a sequence (vn) in 7-~~(~/), converging to u strongly in Ho ’ p ( U ) , such that supp ( vn ) C C U, luI q.e. in U, and 0 q.e. in U. Since u E L~ (U’) for every open set U’ C C U, each function vn belongs to We may also assume that ( vn ) converges to u quasi everywhere in U and thus by Fatou’s lemma

Taking v = vn in (6.17) we get - ( f, vn ) - As n - oo we obtain j~ lulPdJ1 S ( f, u) - (Au, u) +00, and thus u E L~(U). By Proposition 5.5 the functions v E with supp ( v ) cc U are dense in As u E L~(~/), (6.17) implies that u is the solution of problem (6.2) with S2 replaced by U. Since the limit does not depend on the subsequence, the proof is complete. 0 288

COROLLARY 6.13. Let pn, p E M p (Q). Let a family of open subsets of Q which covers Q. Then y A -converges to it in S2 if and only if yA-converges to JL in Qi for every i E I. PROOF. The conclusion follows easily from the compactness of the vergence (Theorem 6.5), from the localization property (Theorem 6.12), and from the uniqueness of the yA-limit (Remark 6.4). D

Acknowledgments. This work was done with the financial support of the Project EURHomogenization, Contract SC1-CT91-0732 of the Program SCIENCE of the Commission of the European Communities, and of the Project "Irregular Variational Problems" of the Italian National Research Council.

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